Question

Find the equation of the line that passes through the point left-parenthesis 1 comma 8 right-parenthesis and is perpendicular to the line negative startfraction 4 over 5 endfraction x plus startfraction 5 over 4 endfraction y equals negative 4.

Answer 1

Don't be shy about using actual parentheses and commas.

Line through (1,8) perpendicular to    

$-\dfrac 4 5 x + \dfrac 5 4 y = -4$

The perpendicular family is gotten by swapping the coefficients on x and y, negating exactly one of them.   The constant is given directly by the point we're through:

$\dfrac 5 4 x + \dfrac 4 5 y = \dfrac 5 4 (1) + \dfrac 4 5 (8) = \dfrac 5 4 + \dfrac{32}{5}$

Let's clear the fractions by multiplying both sides by 20.

$25x + 16y = 25 + 128 = 153$

Might as well stop here.

Answer: 25 x + 16 y = 153